Action dimension of lattices in Euclidean buildings

Abstract

We show that if a discrete group $\mathrm{\Gamma }$ acts properly and cocompactly on an $n$–dimensional, thick, Euclidean building, then $\mathrm{\Gamma }$ cannot act properly on a contractible $\left(2n-1\right)$–manifold. As an application, if $\mathrm{\Gamma }$ is a torsion-free $S$–arithmetic group over a number field, we compute the minimal dimension of contractible manifold that admits a proper $\mathrm{\Gamma }$–action. This partially answers a question of Bestvina, Kapovich, and Kleiner.